Is it permissible for someone who missed the Ishā’ prayer and joined the Imam in the Witr prayer to pray with him and supplicate, then stand up after the Imam’s Taslīm to pray three Rakahs? And is the Rakah he prayed with the Imam considered Witr or part of Ishā’? What is the correct course of action in such a situation?

The 'Isha prayer behind an Imam leading the Tarawih or Witr prayer is valid, according to Imam Al-Shafi'i and Ibn Al-Mundhir. They based their argument on the fact that Mu'adh ibn Jabal used to pray with the Prophet, peace and blessings be upon him, then return and lead his people in the same prayer. Also, the Prophet, peace and blessings be upon him, prayed with one group of his companions during a state of fear, performing two rak'ahs then concluding the prayer, then prayed with another group for two rak'ahs then concluded. The second prayer for the Prophet was supererogatory, yet he led those performing a fard (obligatory) prayer. As for the rak'ah you prayed with the Imam with the intention of 'Isha, it does not count as Witr, because Witr is only valid after completing the 'Isha prayer.